DSQ: Calculating the length of a side of a polygon - GMAT Quantitative Reasoning
Card 0 of 9
True or false: 
is the shortest side of Pentagon 
.
Statement 1: Pentagon 
has perimeter 65.
Statement 2: 
.
True or false:
Statement 1: Pentagon
Statement 2:
Assume both statements are true. If the pentagon has sides of length 12, 13, 13, 13, and 14, with 
the side of length 12, then 
is the shortest side, and the perimeter is
.
On the other hand, if the pentagon has sides of length 11, 12, 13, 14, and 15,
with 
the side of length 12, then 
is not the shortest side, and the perimeter is
.
The two statements together are insufficient.
Assume both statements are true. If the pentagon has sides of length 12, 13, 13, 13, and 14, with
On the other hand, if the pentagon has sides of length 11, 12, 13, 14, and 15,
with
The two statements together are insufficient.
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Note: Figure NOT drawn to scale.
Refer to the above figure. What is the length of 
?
Statement 1: 
Statement 2: 
Note: Figure NOT drawn to scale.
Refer to the above figure. What is the length of
Statement 1:
Statement 2:
To find the length of 
, we can extend 
to meet 
at a point 
to form two rectangles, as seen below:
Statement 1 gives no helpful information, since the length of 
, which is not parallel to 
, has no bearing on that side's length.
If we are given Statement 2 alone, then, as seen in the diagram, 
from segment addition, 
, from Statement 2, and from congruence of opposite sides of a rectangle, 
and 
. Therefore, 
, 
, and 
.
To find the length of
Statement 1 gives no helpful information, since the length of
If we are given Statement 2 alone, then, as seen in the diagram,
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Given a regular hexagon 
, what is the length of 
?
Statement 1: The hexagon is circumscribed by a circle with circumference 
.
Statement 2: 
hs length 12.
Given a regular hexagon
Statement 1: The hexagon is circumscribed by a circle with circumference
Statement 2:
Below is a regular hexagon 
, with its three diameters, its center 
, and its circumscribed circle, which also has center 
.
If Statement 1 is true. then the circle, with circumference , has as its diameter 
, which is 12; this makes the two statements equivalent, so we need only establish that one statement is sufficient or insufficient.
Either way, 
, the radius of the hexagon, is 6. The six triangles that are formed by the sides and diameters of a regular hexagon are all equilateral by symmetry, so each side of the hexagon - in particular, - has length 6.
Below is a regular hexagon
If Statement 1 is true. then the circle, with circumference
Either way,
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Note: Figure NOT drawn to scale
Refer to the above figure. Give the length of 
.
Statement 1: 
Statement 2: 
Note: Figure NOT drawn to scale
Refer to the above figure. Give the length of
Statement 1:
Statement 2:
We can construct perpendicular line segments from 
to 
and from 
to 
as follows:
is the hypotenuse of a right triangle 
, so if we can determine the lengths of 
and 
, we can use the Pythagorean Theorem to determine the length of .
Assume Statement 1 alone. By segment addition, 
. Since 
and 
are opposite sides of a rectangle, 
; similarly, 
. It follows by substitution that 
. Since , it follows that 
, and 
. However, no additional information exists to find 
.
Assume Statement alone. By similar reasoning, 
; since 
, 
, and 
. However, no information exists to find 
.
The two statements put together, however, yield both necessary values: 
and 
. By the Pythagorean Theorem,
.
We can construct perpendicular line segments from
Assume Statement 1 alone. By segment addition,
Assume Statement alone. By similar reasoning,
The two statements put together, however, yield both necessary values:
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Give the length of side 
of Hexagon 
.
Statement 1: 
.
Statement 2: Hexagon 
has perimeter 42.
Give the length of side
Statement 1:
Statement 2: Hexagon
Assume both statements to be true, and examine these two scenarios:
Case 1: The hexagon could have six sides of length 7.
Case 2: The hexagon has four sides of length 7, one of which is 
, one side of length 6, and one side - 
- of length 8.
In both situations, 
and the perimeter of the hexagon is 42:
.
The conditions of both statements would be met in both scenarios, so the two statements together are insufficient.
Assume both statements to be true, and examine these two scenarios:
Case 1: The hexagon could have six sides of length 7.
Case 2: The hexagon has four sides of length 7, one of which is
In both situations,
The conditions of both statements would be met in both scenarios, so the two statements together are insufficient.
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Give the length of side 
of Pentagon 
.
Statement 1: 
Statement 2: 
and 
both have length 10.
Give the length of side
Statement 1:
Statement 2:
Statement 1 alone states that 
is congruent to two other sides, but gives no actual measurements. Statement 2 alone gives the actual measurements of two other segments, but without further information, such as how their lengths relates to that of 
, no information about 
can be inferred.
Now, assume both statements to be true. From Statement 2, 
has length 10, and from Statement 2, 
, which is the same line segment (which can be named after its endpoints in either order), has the same length as 
. Therefore, 
.
Statement 1 alone states that
Now, assume both statements to be true. From Statement 2,
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Give the length of side 
of Pentagon 
.
Statement 1: Pentagon 
has perimeter 50.
Statement 2: 
Give the length of side
Statement 1: Pentagon
Statement 2:
Statement 1 alone only gives the perimeter - the sum of the lengths of the sides - but gives no information about the individual sidelengths. (In particular, there is no indication that the pentagon is regular).
Assume Statement 2 alone. 
and 
are two names for the same line segment, which can be named after its endpoints in either order. Therefore, 
.
Statement 1 alone only gives the perimeter - the sum of the lengths of the sides - but gives no information about the individual sidelengths. (In particular, there is no indication that the pentagon is regular).
Assume Statement 2 alone.
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Note: Figure NOT drawn to scale.
Refer to the above diagram. Evaluate 
.
Statement 1: 
Statement 2: 
Note: Figure NOT drawn to scale.
Refer to the above diagram. Evaluate
Statement 1:
Statement 2:
Refer to the figure below, in which 
has been constructed, and the top and right sides of the figure have been extended to their intersection 
to form Rectangle 
.
Assume Statement 1 alone. Since opposite sides of a rectangle have the same length, 
. By segment addition, 
, and, since , by substitution, 
. Therefore, 
, and the Pythagorean Theorem can be used to find 
:
Since opposite sides of a rectangle have the same length, 
, and by segment addition, 
. By substitution, 
, and 
.
Assume Statement 2 alone. Since 
, the hypotenuse of right triangle 
, and 
, one of its legs, have lengths 15 and 12, respectively, the length of the other leg 
can be found using the Pythagorean Theorem:
Refer to the figure below, in which
Assume Statement 1 alone. Since opposite sides of a rectangle have the same length,
Since opposite sides of a rectangle have the same length,
Assume Statement 2 alone. Since
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True or false: 
is the longest side of Hexagon 
.
Statement 1: 
Statement 2: Hexagon 
has perimeter 66.
True or false:
Statement 1:
Statement 2: Hexagon
Statement 1 alone only gives information about one side of the hexagon, and Statement 2 gives only information about the perimeter without giving any clues as to the individual sidelengths; neither is sufficient to answer the question.
Assume both statements are true. If 
, with length 10, is the longest side of Hexagon 
, then
By the addition property of inequality,
This means the sum of the sidelengths of the hexagon, which is its perimeter, is less than 66, in contradiction to Statement 2. 
cannot be the longest side.
Statement 1 alone only gives information about one side of the hexagon, and Statement 2 gives only information about the perimeter without giving any clues as to the individual sidelengths; neither is sufficient to answer the question.
Assume both statements are true. If
By the addition property of inequality,
This means the sum of the sidelengths of the hexagon, which is its perimeter, is less than 66, in contradiction to Statement 2.
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